Design of Quick-Returning R-S-S-R Mechanisms

1988 ◽  
Vol 110 (4) ◽  
pp. 423-428 ◽  
Author(s):  
F. O. Suareo ◽  
K. C. Gupta
Keyword(s):  
Solution Space ◽  
Algebraic Method ◽  
Axial Distance ◽  
Link Length ◽  
Planar Case ◽  
Transmission Angle ◽  
Crank Angle ◽  
Shaft Angle ◽  
The Given ◽  
Crank Length

An algebraic method is presented to synthesize quick-returning R-S-S-R mechanisms which satisfy the given time-ratio and follower oscillation angle requirements. In these designs, the three parameters, which define the follower spheric joint, satisfy a quadratic condition. When the shaft angle between the input and output shafts is zero, this quadratic condition reduces to the equation of a circle which is a familiar classical result for the planar case. The solution space for the quick-returning R-S-S-R linkage is such that, for each set of choices for crank length a2, follower axial distance S4, and initial follower angle φ0, there are four sets of follower length a4, initial crank angle θ0, crank axial distance S2, and coupler length a3. These designs are screened so that they do not have branch defect, have transmission angle values in a given range, and have reasonable link length proportions.

A Generalized Performance Sensitivity Synthesis Methodology for Four-Bar Mechanisms

1992 ◽  
Author(s):  
Ming-Yih Lee ◽  
Arthur G. Erdman ◽  
Salaheddine Faik
Keyword(s):  
Solution Space ◽  
Design Solution ◽  
Link Length ◽  
Performance Sensitivity ◽  
Closed Chain ◽  
Synthesis Methodology ◽  
Position Synthesis ◽  
The Relationship ◽  
Accuracy Performance ◽  
Sensitivity Maps

Abstract A generalized accuracy performance synthesis methodology for planar closed chain mechanisms is proposed. The relationship between the sensitivity to variations of link lengths and the location of the moving pivots of four-link mechanisms is investigated for the particular objective of three and four position synthesis. In the three design positions case, sensitivity maps with isosensitivity curves plotted in the design solution space allow the designer to synthesize a planar mechanism with desired sensitivity value or to optimize sensitivity from a set of acceptable design solutions. In the case of four design positions, segments of the Burmester design curves that exhibit specified sensitivity to link length tolerance are identified. A performance sensitivity criterion is used as a convenient and a useful way of discriminating between many possible solutions to a given synthesis problem.

Analytic program derivation in type theory

1998 ◽  
Author(s):  
Petri Mäenpää
Keyword(s):  
Mathematical Method ◽  
Problem Solving ◽  
Type Theory ◽  
Mathematical Problem ◽  
Geometric Analysis ◽  
Algebraic Method ◽  
Point Of View ◽  
Program Derivation ◽  
Mathematical Problems ◽  
The Given

This work proposes a new method of deriving programs from their specifications in constructive type theory: the method of analysis-synthesis. It is new as a mathematical method only in the area of programming methodology, as it is modelled upon the most successful and widespread method in the history of exact sciences. The method of analysis-synthesis, also known as the method of analysis, was devised by Ancient Greek mathematicians for solving geometric construction problems with ruler and compass. Its most important subsequent elaboration is Descartes’s algebraic method of analysis, which pervades all exact sciences today. The present work expands this method further into one that aims at systematizing program derivation in a heuristically useful way, analogously to the way Descartes’s method systematized the solution of geometric and arithmetical problems. To illustrate the method, we derive the Boyer-Moore algorithm for finding an element that has a majority of occurrences in a given list. It turns out that solving programming problems need not be too different from solving mathematical problems in general. This point of view has been emphasized in particular by Martin-Löf (1982) and Dijkstra (1986). The idea of a logic of problem solving originates in Kolmogorov (1932). We aim to refine the analogy between programming and mathematical problem solving by investigating the mathematical method of analysis in the context of programming. The central idea of the analytic method, in modern terms, is to analyze the functional dependencies between the constituents of a geometric configuration. The aim is to determine how the sought constituents depend on the given ones. A Greek analysis starts by drawing a diagram with the sought constructions drawn on the given ones, in the relation required by the problem specification. Then the sought constituents of the configuration are determined in terms of the given ones. Analysis was the Greeks’ method of discovering solutions to problems. Their method of justification was synthesis, which cast analysis into standard deductive form. First it constructed the sought objects from the given ones, and then demonstrated that they relate as required to the given ones. In his Geometry, Descartes developed Greek geometric analysis-synthesis into the modern algebraic method of analysis.

Link Length Bounds on the Four-Bar Chain

1971 ◽  
Vol 93 (1) ◽  
pp. 287-293 ◽  
Author(s):  
P. W. Eschenbach ◽  
D. Tesar
Keyword(s):  
Algebraic Curve ◽  
Inequality Constraints ◽  
Kinematic Chain ◽  
Bounded Region ◽  
Practical Application ◽  
Link Length ◽  
Chain Link ◽  
Transmission Angle

The four-link kinematic chain is studied in an effort to establish a bounded region to limit the chain link lengths based upon transmission angle inequality constraints. The resulting constraint limitations are displayed graphically with the chain and their algebraic curve properties are delineated. Simple approximations of these complex loci are developed to facilitate practical application.

MATHEMATICAL MODEL AND AREA OF EXISTENCE OF THE FAMILY CRANKS AND MOBILE MECHANISMS WITH THE MAXIMUM TRANSMISSION ANGLE

2020 ◽  
pp. 20-26
Author(s):  
N. A. Sereda
Keyword(s):  
Mathematical Model ◽  
New Family ◽  
Arm Span ◽  
Angle Function ◽  
Straight Line ◽  
Transmission Angle ◽  
Crank Angle ◽  
The Family ◽  
Unit Radius ◽  
Maximum Transmission

The article examines crank-rocker mechanisms. Such mechanisms are used in transport and technological machines. The article is devoted to the search for a new family of crank-rocker mechanisms. A mathematical model of a new family of crank-rocker mechanisms is obtained. In this family, the maximum transmission angle reaches 90 when the crank angle is 75. Thus, the new family of crank-rocker mechanisms differs from the known families by the position of the mechanism in which the maximum of the transmission angle function takes place. It is shown that, with a certain ratio of link lengths, the new family corresponds to the known families KKM-5 and KKM-7. The area of existence of a new family of crank-rocker mechanisms is established. This area is bounded by the arc of the circle of the unit radius and the curve. The mentioned curve is based on a joint solution of a mathematical model of a new family of mechanisms and the famous Kolchin straight line. The dependence for the minimum transmission angle is obtained. A formula for determining the angle of the rocker arm span is proposed. A graphical interpretation of the mentioned dependencies and formulas is constructed. The scope of existence of a new family of crank-rocker mechanisms and graphical interpretations are used in determining the geometric parameters of mechanisms. These mechanisms are part of a new family of mechanisms.

Conceptual Design Method Based on Behavior Re-Creation

2007 ◽  
Vol 10-12 ◽  
pp. 198-202
Author(s):  
H.B. Miao
Keyword(s):  
Conceptual Design ◽  
Process Model ◽  
Design Method ◽  
Solution Space ◽  
The Novel ◽  
Creation Process ◽  
Design Scheme ◽  
Mechanical Products ◽  
Basic Behavior ◽  
The Given

Conceptual design for the mechanical products is the most important and complex phase. Most of the human’s creativity is exhibited in this phase. So many researchers in the world have made researches on its theory and methodology. So far there exists an evident limitation for the modern conceptual design methodology. That is to say, the design scheme obtained form conceptual design is only one solution to the given design problem, which cannot guarantee the creativity and novelty of the design scheme. In order to improve the creativity of conceptual design, the re-creation process model for the intelligent conceptual design is presented in this paper based on function-behavior-structure model. This model expanded the problem’s solution space by degrading the space of behavior and granularity, and describing the behavior space using the more basic behavior. The key of re-creation process model is the transformation of behavior and granularity space. Taking the theory of quotient space as the math description tool, the transformation for the space of behavior and granularity is studied in detail in this paper. At last, an example is given to prove that it is easy to obtain the novel and creative design scheme applying the method presented in this paper.

A Higher-Order Analysis of Basic Linkages for Harmonic Motion Generation

1987 ◽  
Vol 109 (3) ◽  
pp. 301-307 ◽  
Author(s):  
K. Farhang ◽  
A. Midha ◽  
A. Bajaj
Keyword(s):  
Perturbation Technique ◽  
Motion Generation ◽  
Link Length ◽  
Small Motion ◽  
Harmonic Motion ◽  
Order Analysis ◽  
Modified Equations ◽  
Crank Angle ◽  
The Mean ◽  
Definition Of

In an earlier work, a perturbation technique was first presented to obtain approximate simple harmonic equations for describing the output motions of rudimentary linkages, i.e., a crank-rocker and a slider-crank, with relativley small input cranks. The technique involved consideration of a small motion excursion about a so-called “mean linkage configuration.” These equations were facilitated through truncation of the binomial series expansion of the output motions, expressed in terms of the input crank angle. Assuming a small crank to ground link length ratio, terms containing second or higher powers of this ratio were neglected. This paper retains terms containing higher powers in an effort to improve upon (i) the definition of the mean linkage configuration, and (ii) the harmonic motion content representation of the output motions. The improvements made due to the “modified” equations, relative to the “original” ones, are pictorially presented as being significant.

Design of Centric Drag-Link Mechanisms for Delay Generation With Focus on Space Occupation

2008 ◽  
Vol 131 (1) ◽  
Author(s):  
Abdullah F. Al-Dwairi
Keyword(s):  
Length Ratio ◽  
Link Length ◽  
Link Mechanism ◽  
Maximum Delay ◽  
Transmission Angle ◽  
Quality Of Motion ◽  
Nonuniform Rotation ◽  
Drag Link ◽  
Space Occupation

Planar drag-link mechanism is a Grashofian four-bar chain with the shortest link fixed. In practice, the mechanism is used as a coupling between two shafts to convert uniform rotation of the driving shaft into a nonuniform rotation of the driven shaft. The nonuniformity in rotation is characterized by a cyclically increasing and decreasing delay (or advance) in the displacement of the driven shaft relative to that of the driving shaft. Drag-link synthesis problems include synthesizing the mechanism to generate a specified maximum delay. In a drag-link mechanism, the longer links make a full rotation about fixed pivots, which results in a relatively large installation space. This calls for designing drag-link mechanisms with a focus on space occupation, along with the traditional criteria of quality of motion transmission. Using position analysis, we investigate the relationships among mechanism space occupation, extreme transmission angle, and the generated maximum delay. Space occupation is represented by the link-length ratio of input link to fixed link. Given a desired maximum delay, the proposed approach suggests finding a unique extreme transmission angle value for which this link-length ratio is at a minimum. A closed-form solution to drag-link synthesis to generate a specified maximum delay is developed based on a compromise between quality of motion transmission and space occupation. For any drag-link designed by this compromise, the coupler link and the output crank are of the same length. Based on the obtained design equations, a graphical design solution and a method for evaluating space occupation are provided.

scholarly journals Special Orthogonal Polynomials in Quantum Mechanics

2019 ◽  
Author(s):  
Abdulaziz D. Alhaidari
Keyword(s):  
Differential Equation ◽  
Quantum Mechanics ◽  
Orthogonal Polynomials ◽  
Solution Space ◽  
Type Equation ◽  
Algebraic Method ◽  
Recursion Relations ◽  
New Class ◽  
Continuous And Discrete Spectra ◽  
Discrete Spectra

Using an algebraic method for solving the wave equation in quantum mechanics, we encountered a new class of orthogonal polynomials on the real line. One of these is a four-parameter polynomial with a discrete spectrum. Another that appeared while solving a Heun-type equation has a mix of continuous and discrete spectra. Based on these results and on our recent study of the solution space of an ordinary differential equation of the second kind with four singular points, we introduce a modification of the hypergeometric polynomials in the Askey scheme. Up to now, all of these polynomials are defined only by their three-term recursion relations and initial values. However, their other properties like the weight function, generating function, orthogonality, Rodrigues-type formula, etc. are yet to be derived analytically. This is an open problem in orthogonal polynomials.

Optimum Design of the Crank Rocker Four Bar Mechanism for Specified Output Oscillation Angle and Timing Ratio

1987 ◽  
Author(s):  
D. H. Suchora ◽  
G. Wrightson
Keyword(s):  
Optimum Design ◽  
Cycle Time ◽  
Geometric Construction ◽  
Length Ratio ◽  
Total Output ◽  
Link Length ◽  
Analytic Construction ◽  
Transmission Angle ◽  
Work First ◽  
Parameters Of Interest

Abstract In designing a crank rocker four bar mechanism with a uniform input rotation typical input parameters are the required total output oscillation angle and the timing ratio of the advance to return cycle time. In determining an optimum design the parameters of interest are usually the extreme transmission angles and the ratio of the longest to shortest link length occuring in the mechanism. This work first develops an analytic construction of the link lengths and worst transmission angles based on the necessary geometry for a given output angle of oscillation and required timing ratio. The resulting equations are programmed and graphs developed which give the variation of extreme transmission angle and maximum link length ratio as a function of the specified output angle of oscillation, timing ratio, and geometric construction variables. Using these graphs a designer will be able to easily select optimum designs based on worst transmission angles and link length ratios. Examples are included.

scholarly journals Translation groups of the boundary-layer flows induced by continuous moving surfaces

2010 ◽  
Vol 655 ◽  
pp. 327-343 ◽  
Author(s):  
EUGEN MAGYARI
Keyword(s):  
Boundary Layer ◽  
Solution Space ◽  
Outer Edge ◽  
Plane Boundary ◽  
Continuous Group ◽  
Balance Equations ◽  
Boundary Layer Flows ◽  
Layer Flow ◽  
Steady Plane ◽  
The Given

The steady plane boundary-layer flows of velocity field {u(x, y), v(x, y)} induced by continuous moving surfaces are revisited in this paper. It is shown that the governing balance equations, as well as the asymptotic condition u(x, ∞) = 0 at the outer edge of the boundary layer are invariant under arbitrary translations y → y + y0(x) of the transverse coordinate y. The wall conditions, i.e. the prescribed stretching velocity u(x, 0) ≡ Uw(x) and the transpiration velocity v(x, 0) ≡ Vw(x) distributions, however, undergo in general substantial changes. The consequences of this basic symmetry property on the structure of the solution space are investigated. It is found that starting with a primary solution which describes the boundary-layer flow induced by an impermeable surface, infinitely many translated solutions can be generated which form a continuous group, the translation group of the given primary solution. The elements of this group describe boundary-layer flows induced by permeable surfaces stretching under transformed wall conditions, Uw(x) → Ũw(x) = u[x, y0(x)] and Vw(x) → Ṽw(x) = v[x, y0(x)] − y′0(x)u[x, y0(x)], respectively. In this way, starting with a known solution {u(x, y), v(x, y)} so that the inverse y0(x) = u−1(x, Ũw) of u[x, y0(x)] exists, a new solution {ũ(x, y), ṽ(x, y)} corresponding to any desired stretching velocity distribution Ũw(x) can be prepared. It also turns out that several exact solutions discovered during the latter decades are not basically new solutions, but translated counterparts of some formerly reported primary solutions. A few specific examples are discussed in detail.

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